Simplified version - In Parallel, you add LED Current. In Series, you add LED Voltage. On top of that, if you need LEDs individually addressable, Parallel is the only way, since Series LEDs are tied together to act as one . Use whatever's appropriate for your setup.
Using my Crossguard as an example, I had five 2.7v LEDs to run. Three at 700ma+ each and two at ~500mA each. at roughly 800mA each.
In parallel, I'd need a >2.7v battery that could pump out 4-5A - which a single 18650 can easily do with some small resistors.
In series, the main star would be acting as an 8.1v 700mA+ LED with the combined crossguards as a 5.4v LED at about 500mA. So I'd need a >8.1v battery that could support 1.5-2A, which brings me to a 3x 14500 pack and some beefy resistors.
Parallel is the obvious choice there.
On the flip side, Graflex needed a BrBW star and used a soundboard that needed 5v+ for full-volume audio.
If I wanted to accept the volume loss, I could've settled for an 18650 and wired all the LEDs in parallel with tiny resistors. But I didn't want to lose power.
Parallel wasn't even an option for the 2x 14500 pack I was using. All three LEDs together would've pulled ~3A, and 14500s can only put out ~2A.
Solution was to get the main blade LEDs in series, or replace the battery pack.
So the two blues were wired in series to make a ~7v 1000mA LED with white running parallel at about 3.1v and 1000mA. Blues used a tiny resistor, white needed a beefy one, but the star only pulled about 2A during Flash.
As for bridging everything on the star, I used to do it with LED-Supply tri-stars, since the center holes on the star and heatsink could only fit 4 wires max, but it was a real pain having to criss-cross wires. Now I just drill those center holes out to 3/16" to fit all six 26ga wire. Makes things a lot easier on me.
On SF, SH, and TCSS heatsinks, it's a non-issue, since you can use the holes on the perimeter of the star and heatsink.
SF stars are already designed with a bridged positive, and negatives are easy to bridge and often come that way.